
A ellipse is a closed curve that can be represented by the equation if the major axis is horizontal, or if the major axis is vertical.^{[1]}Something that is related to an ellipse or is in the shape of an ellipse can be called elliptic or elliptical. For example, elliptic geometry is a geometry that can be visualized as taking place on the surface of an ellipse. 
Variable or Parameter  Description 

x  The variable x represents the horizontal axis. 
y  The variable y represents the vertical axis. 
a  The parameter a represents the length of the semimajor axis. 
b  The parameter b represents the length of the semiminor axis. 
h  The parameter h represents the x position of the center of the ellipse, which is at the intersection of the major axis and minor axis. 
k  The parameter k represents the y position of the center of the ellipse, which is at the intersection of the major axis and minor axis. 
Table 1 
Each ellipse has two axes, a major axis and a minor axis. Each axis is a line around which the ellipse is symmetrical. One is horizontal and the other is vertical. The major axis of an ellipse is the larger of the two axes. The major axis is also called the principal axis. The minor axis of an ellipse is the smaller of the two axes. Half the major axis is called the semimajor axis. Half the minor axis is called the semiminor axis. In the equation for the ellipse, a is the length of the semimajor axis, and b is the length of the semiminor axis. (h,k) is a point at the center of the ellipse. Changing h moves the ellipse to the left or right. Changing k moves the ellipse up or down. An ellipse is a conic section formed by intersecting a plane with a cone. 
Manipulative 1: Ellipse. Click for classroom demonstrator. Created with GeoGebra. 
The endpoints of the horizontal axis are at . The endpoints of the horizontal ellipse are at . In manipulative 1, click on the 'Show major axis' to see the major axis and click one the 'show minor axis' to see the minor axis. Click on the blue points on the sliders in manipulative 1 and drag them to change the ellipse.
Each ellipse has two foci. Each focus can be found on the major axis. for a horizontal ellipse, or where . The foci are two points on the major axis of the ellipse. For each point in the ellipse, the sum of the distance of that point to the foci is equal to 2a. Click on the 'Show foci' checkbox in manipulative 1 to show the foci. Click on the blue point on the ellipse and drag it to change the figure.
The eccentricity of an ellipse is a measure of how much it is changed from a circle. In the case where b=a, the ellipse is a circle and the eccentricity is 0. The formula for the eccentricity of an ellipse is
where b<a. Another form of the equation of the ellipse uses the length of the semimajor axis and the eccentricity:An ellipse can also be defined with one focus point and a directrix. An ellipse is all points such that the ratio of the distance from the focus to the point on the ellipse divided by the distance from the point on the ellipse to the directrix is equal to the ratio of c to a.
Name  Formula 

Equation of an ellipse if the major axis is horizontal.  
Equation of an ellipse if the major axis is horizontal.  
Endpoints of the horizontal axis of a horizontal ellipse.  where c^{2} = a^{2}  b^{2} 
Endpoints of the vertical axis of a vertical ellipse.  where c^{2} = a^{2}  b^{2} 
Foci of a horizontal ellipse.  where c^{2} = a^{2}  b^{2} 
Foci of a vertical ellipse.  where c^{2} = a^{2}  b^{2} 
Eccentricity of an ellipse.  
Area of an ellipse.  where a and b are the lengths of the major and minor axes of the ellipse. 
The circumference of an ellipse is expressed using an infinite series.  where ε is the eccentricity. 
Table 2  Formulas related to an ellipse. 
The general form of the elliptical equation is
Step  Example  Description 

1  First, plot the center of the ellipse at
. In this example
.
For the xcoordinate, start with . Now subtract x from both sides to get . Multiply both sides by 1 to get . For the ycoordinate, start with . Now subtract y from both sides to get . Multiply both sides by 1 to get .  
2  Now plot the endpoints of the horizontal axis. The left endpoint is at . In this case that is . Plot the right endpoint at . For this equation, .  
3  Now plot the endpoints of the horizontal axis. The bottom endpoint is at . In this case that is . Plot the top endpoint at . For this equation, .  
4  Now sketch the ellipse.  
5  To plot the foci, calculate the value of c. The equation for c is . Using the values of a and b, , and . Since the major axis is the horizontal axis, the foci are at . Substituting the values gives and .  
Table 2 
#  A  B  C  D 
E  F  G  H  I 
J  K  L  M  N 
O  P  Q  R  S 
T  U  V  W  X 
Y  Z 
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